Abstract
The Fateev-Zamolodchikov-Zamolodchikov (FZZ) duality relates Witten’s cigar model to sine-Liouville theory. This duality was proven in the path integral formulation and extended to the case of higher genus closed Riemann surfaces by Schomerus and one of the authors. In this note we further extend the duality to the case with boundary. Specifically, we relate D1-branes in the cigar model to D2-branes in the sine-Liouville theory. In particular, the boundary action for D2-branes in the sine-Liouville theory is constructed. We also consider the fermionic version of the FZZ duality. This duality was proven as a mirror symmetry by Hori and Kapustin, but we give an alternative proof in the path integral formulation which directly relates correlation functions. Also here the case with boundary is investigated and the results are consistent with those for branes in \( \mathcal{N} = 2 \) super Liouville field theory obtained by Hosomichi.
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References
V. Fateev, A.B. Zamolodchikov and A.B. Zamolodchikov, unpublished.
E. Witten, On string theory and black holes, Phys. Rev. D 44 (1991) 314 [SPIRES].
K. Hori and A. Kapustin, Duality of the fermionic 2d black hole and \( \mathcal{N} = 2 \) Liouville theory as mirror symmetry, JHEP 08 (2001) 045 [hep-th/0104202] [SPIRES].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [SPIRES].
V. Kazakov, I.K. Kostov and D. Kutasov, A matrix model for the two-dimensional black hole, Nucl. Phys. B 622 (2002) 141 [hep-th/0101011] [SPIRES].
A. Giveon and D. Kutasov, Little string theory in a double scaling limit, JHEP 10 (1999) 034 [hep-th/9909110] [SPIRES].
Y. Hikida and V. Schomerus, The FZZ-duality conjecture — A proof, JHEP 03 (2009) 095 [arXiv:0805.3931] [SPIRES].
K. Gawędzki and A. Kupiainen, Coset construction from functional integrals, Nucl. Phys. B 320 (1989) 625 [SPIRES].
S. Ribault and J. Teschner, H +3 WZNW correlators from Liouville theory, JHEP 06 (2005) 014 [hep-th/0502048] [SPIRES].
Y. Hikida and V. Schomerus, H +3 WZNW model from Liouville field theory, JHEP 10 (2007) 064 [arXiv:0706.1030] [SPIRES].
J.P. Babaro and G. Giribet, Disk one-point function for non-rational conformal theories, JHEP 09 (2010) 077 [arXiv:1005.2607] [SPIRES].
T. Creutzig and Y. Hikida, Branes in the OSP(1–2) WZNW model, Nucl. Phys. B 842 (2011) 172 [arXiv:1004.1977] [SPIRES].
T. Creutzig, Branes in supergroups, arXiv:0908.1816 [SPIRES].
T. Creutzig and P.B. Ronne, The GL(1–1)-symplectic fermion correspondence, Nucl. Phys. B 815 (2009) 95 [arXiv:0812.2835] [SPIRES].
T. Creutzig, Geometry of branes on supergroups, Nucl. Phys. B 812 (2009) 301 [arXiv:0809.0468] [SPIRES].
T. Creutzig and V. Schomerus, Boundary correlators in supergroup WZNW models, Nucl. Phys. B 807 (2009) 471 [arXiv:0804.3469] [SPIRES].
T. Creutzig, T. Quella and V. Schomerus, Branes in the GL(1–1) WZNW -model, Nucl. Phys. B 792 (2008) 257 [arXiv:0708.0583] [SPIRES].
T. Creutzig and P.B. Ronne, From world-sheet supersymmetry to super target spaces, JHEP 11 (2010) 021 [arXiv:1006.5874] [SPIRES].
V. Fateev and S. Ribault, Boundary action of the H +3 model, JHEP 02 (2008) 024 [arXiv:0710.2093] [SPIRES].
V. Fateev, A.B. Zamolodchikov and A.B. Zamolodchikov, Boundary Liouville field theory. I: Boundary state and boundary two-point function, hep-th/0001012 [SPIRES].
J. Teschner, Remarks on Liouville theory with boundary, hep-th/0009138 [SPIRES].
S. Ribault and V. Schomerus, Branes in the 2D black hole, JHEP 02 (2004) 019 [hep-th/0310024] [SPIRES].
A.Y. Alekseev and V. Schomerus, D-branes in the WZW model, Phys. Rev. D 60 (1999) 061901 [hep-th/9812193] [SPIRES].
K. Hosomichi and S. Ribault, Solution of the H +3 model on a disc, JHEP 01 (2007) 057 [hep-th/0610117] [SPIRES].
A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [SPIRES].
K. Hosomichi, \( \mathcal{N} = 2 \) Liouville theory with boundary, JHEP 12 (2006) 061 [hep-th/0408172] [SPIRES].
I. Bakas and E. Kiritsis, Beyond the large-N limit: Nonlinear W ∞ as symmetry of the \( {{{{\text{SL}}\left( {2,\mathbb{R}} \right)}} \left/ {{{\text{U}}(1)}} \right.} \) coset model, Int. J. Mod. Phys. A 7S1A (1992) 55 [hep-th/9109029] [SPIRES].
V.A. Fateev and S.L. Lukyanov, Boundary RG flow associated with the AKNS soliton hierarchy, J. Phys. A 39 (2006) 12889 [hep-th/0510271] [SPIRES].
K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [SPIRES].
T. Eguchi and Y. Sugawara, Modular bootstrap for boundary \( \mathcal{N} = 2 \) Liouville theory, JHEP 01 (2004) 025 [hep-th/0311141] [SPIRES].
C. Ahn, M. Stanishkov and M. Yamamoto, One-point functions of \( \mathcal{N} = 2 \) super-Liouville theory with boundary, Nucl. Phys. B 683 (2004) 177 [hep-th/0311169] [SPIRES].
D. Israel, A. Pakman and J. Troost, D-branes in \( \mathcal{N} = 2 \) Liouville theory and its mirror, Nucl. Phys. B 710 (2005) 529 [hep-th/0405259] [SPIRES].
A. Fotopoulos, V. Niarchos and N. Prezas, D-branes and extended characters in \( {{{{\text{SL}}\left( {2,\mathbb{R}} \right)}} \left/ {{{\text{U}}(1)}} \right.} \), Nucl. Phys. B 710 (2005) 309 [hep-th/0406017] [SPIRES].
Y. Kazama and H. Suzuki, New \( \mathcal{N} = 2 \) Superconformal field theories and superstring compactification, Nucl. Phys. B 321 (1989) 232 [SPIRES].
A. Giveon, A. Konechny, A. Pakman and A. Sever, Type 0 strings in a 2D black hole, JHEP 10 (2003) 025 [hep-th/0309056] [SPIRES].
J.M. Maldacena, Long strings in two dimensional string theory and non-singlets in the matrix model, JHEP 09 (2005) 078 [Int. J. Geom. Meth. Mod. Phys. 3 (2006) 1] [hep-th/0503112] [SPIRES].
O.D. Andreev and A.A. Tseytlin, Partition function representation for the open superstring effective action: cancellation of Mobius infinities and derivative corrections to Born-Infeld Lagrangian, Nucl. Phys. B 311 (1988) 205 [SPIRES].
A.A. Tseytlin, σ-model approach to string theory effective actions with tachyons, J. Math. Phys. 42 (2001) 2854 [hep-th/0011033] [SPIRES].
P. Kraus and F. Larsen, Boundary string field theory of the DD-bar system, Phys. Rev. D 63 (2001) 106004 [hep-th/0012198] [SPIRES].
T. Takayanagi, S. Terashima and T. Uesugi, Brane-antibrane action from boundary string field theory, JHEP 03 (2001) 019 [hep-th/0012210] [SPIRES].
A.B. Zamolodchikov and A.B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0101152 [SPIRES].
G. Giribet, Y. Hikida and T. Takayanagi, Topological string on OSP(1–2)/U(1), JHEP 09 (2009) 001 [arXiv:0907.3832][SPIRES].
Y. Hikida and V. Schomerus, Structure constants of the OSP(1–2) WZNW model, JHEP 12 (2007) 100 [arXiv:0711.0338] [SPIRES].
T. Creutzig, Y. Hikida and P.B. Ronne, Supergroup — extended super Liouville correspondence, JHEP 06 (2011) 063 [arXiv:1103.5753] [SPIRES].
M.A. Bershadsky, Superconformal algebras in two-dimensions with arbitrary N, Phys. Lett. B 174 (1986) 285 [SPIRES].
V.G. Knizhnik, Superconformal algebras in two-dimensions, Theor. Math. Phys. 66 (1986) 68 [Teor. Mat. Fiz. 66 (1986) 102] [SPIRES].
T. Creutzig, P.B. Ronne and V. Schomerus, \( \mathcal{N} = 2 \) superconformal symmetry in super coset models, Phys. Rev. D 80 (2009) 066010 [arXiv:0907.3902] [SPIRES].
K. Gawędzki, Boundary WZW, G/H, G/G and CS theories, Annales Henri Poincaré 3 (2002) 847 [hep-th/0108044] [SPIRES].
S. Fredenhagen and V. Schomerus, D-branes in coset models, JHEP 02 (2002) 005 [hep-th/0111189] [SPIRES].
S. Ribault, Knizhnik-Zamolodchikov equations and spectral flow in AdS 3 string theory, JHEP 09 (2005) 045 [hep-th/0507114] [SPIRES].
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ArXiv ePrint:1012.4731
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Creutzig, T., Hikida, Y. & Rønne, P.B. The FZZ duality with boundary. J. High Energ. Phys. 2011, 4 (2011). https://doi.org/10.1007/JHEP09(2011)004
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DOI: https://doi.org/10.1007/JHEP09(2011)004